We briefly summarise our work highlighting the contributions. After introducing
the basic definitions and notation we recall the notion of a \(K\) -weighted
automaton (Section REF ) and and \(K\) -weighted bisimulation
(Section REF ). We use the definition of Bonchi et al. [1]} but assume that the underlying set of weights is a
semiring \(K\) . We use the terms \(K\) -weighted automaton and \(K\) -weighted
bisimulation to distinguish our setting from that of
[1]}. The fact that \(K\) -weighted automata are
\(\mathcal {W}\) -coalgebras for an endofunctor \(\mathcal {W}\)  on \(\mathsf {Set}\)  follows, and so does
the correspondence between \(K\) -weighted bisimulations and kernels of
\(\mathcal {W}\) -homomorphims (Section REF ). This result uses our definition of
kernels, so there is some novelty here. The proof that \(K\) -weighted bisimilarity \(\sim _w\) 
coincides with \(\mathcal {W}\) -behavioural equivalence \(\approx _\mathcal {W}\)  is the one
in [1]} presented in a different way.
